On involutes of order k of a null Cartan curve in Minkowski spaces

Abstract

In this paper, we define an involute and an evolving involute of order $k$ of a null Cartan curve in Minkowski space $\mathbb{E}^{n}_{1}$ for $n\ge 3$ and $1\le k\le n-1$. In relation to that, we prove that if a null
Cartan helix has a null Cartan involute of order 1 or 2, then it is Bertrand null Cartan curve and its involute
is its Bertrand mate curve. In particular, we show that Bertrand mate curve of Bertrand null Cartan curve can also be a
non-null curve and find the relationship between the Cartan frame of a null Cartan curve and the Frenet or the Cartan frame of its non-null or null Cartan involute of order $1\le k\le 2$. We show that among all null Cartan curves in $\mathbb{E}^3_1$, only the null Cartan cubic has two families of involutes of order $1$, one of which lies on $B$-scroll. We also give some relations between involutes of orders $1$ and $2$ of a null Cartan curve in Minkowski $3$-space. As an application, we show that involutes of order $1$ of a null Cartan curve in $\mathbb{E}^3_1$, evolving according to null Betchov-Da Rios vortex filament equation, generate timelike Hasimoto surfaces.